AI 中文总结
本文研究将有限群概形实现为光滑射影簇基本群概形的问题，通过贝尔蒂尼型结果、戈多-塞雷构造等方法，证明在无限完美域k上任意有限群概形G/k均可作为对应类型的基本群概形实现于特定维数的连通光滑射影簇。
AI 中文摘要
本文研究将有限群概形实现为光滑射影簇基本群概形的问题。我们建立了贝尔蒂尼型结果，并证明了射影空间中轻度奇异完全交的基本群概形是平凡的。基于此，我们运用戈多-塞雷构造证明：在无限完美域k上，任意有限群概形G/k都可实现为连通光滑射影簇的S-基本群概形、扩展诺里基本群概形及诺里基本群概形，该簇的维数至少为dim Lie(G)+2。
英文摘要
In this paper we study the problem of realizing finite group schemes as fundamental group schemes of smooth projective varieties. We establish Bertini-type results and prove the triviality of fundamental group schemes of mildly singular complete intersections in projective space. With this in hand, we are able to go through a Godeaux--Serre construction and prove that, over an infinite perfect field $k$, any finite group scheme $G/k$ can be realized as the $S$-, extended Nori and Nori fundamental group schemes of a connected smooth projective variety of any dimension at least $\dim \text{Lie}(G)+2$.
Comments21 pages, comments are welcome!
